 Okay, good morning everybody. So today I will continue speaking about Quasi crystals. And there are two things I'd like to do. The first thing is to prove this theorem about the density of a Quasi crystal. And the Quasi crystal here always means the Quasi crystal of cat and project type such a set which Jens described yesterday. So we want to prove how many points are there in such a Quasi crystal in a large region. And that proof will also allow me to illustrate the technique which is very important in applications of dynamics in number theory. So there are a number of applications where say Rattner's theorem or other equidistribution theorems in homogeneous dynamics have been applied to counting integer solutions to some Diophantine equation or solving Diophantine problems. So count how many integer points are there near some manifold or some variety in RM. And how, so it's a basic question. How can such a question be addressed using equidistribution of some orbit? Well the trick is you kind of thicken the integer points a little bit and then you are able to count such solutions. And the second thing I wanted to do is to discuss visible points in the lattice. So this is what Jens talked about at the end of the talk. So we'll come to that. But, so I will remind you first of some definitions. And I hope you have Jens and the notes from yesterday in front of you. So the picture, the first, I will try to draw some pictures as well. But, so we are going to define this catam project set in RD. And in order to do so, we extend RD by some internal dimensions. So we consider the product space, Rn equals RD times RM. And then we introduced the notations for these orthogonal projections. So we call pi the projection from Rn to the physical space RD. And we call pi interior, the projection from Rn to the interior space of the, what is it called? Model space RM. Okay, and then in Rn, we fix a lattice L. And before defining a quasi-crystal, we made some assumptions. We assumed that the, if I look at the projection of all the lattice points to the model space RM, or the interior space RM, we assume that, okay, this is always some, if I take the closure of this, it's always some close, stability and subgroup of RM. We called this closure A. But today, just for simplicity of notation, we'll assume that actually this is equal to all of RM. There are no extra difficulties. The fact that we are, this is a standard assumption to assume that we are dense in all of RM here. And the fact that in Jens and my work, we want to have a slightly more general setting, that is just for convenience. It allows us to cover a few more examples and so on. So just to refer back to Jens talk yesterday, this means that A and also the connected component of A, containing the origin, both of these are equal to RM and also the V, which was defined as the product of RD and RM. That is now equal to all of RM. So that you can refer back to this. And now we fix also a regular window set. So we'll let W be a subset of RM. In general, it's a subset of A. We'll let this be a regular window set. And recall, this means that the boundary of W has finite Lebesgue measure. And it has non-empty interior and it is a bounded set. So this means that W is bounded. It has non-empty interior and the Lebesgue measure of the boundary of W is zero. So W is a Jordan measurable set if you want. And then finally, we define, maybe I'll do it here. The quasi-crystal as the set of all points that you get, you run through all the points Y in the lattice L such that the interior project, projection of Y is in the window. So we run through all the lattice points which project into the window and we consider the projection of these two physical space. That's the quasi-crystal. And this is our fixed set P. And there wasn't one extra assumption. We assume also that any point in here, any point in this quasi-crystal only has a single representative in the lattice. So there is a certain injectivity. So we also assume that for any point in X, this was, yeah, this also Jens did this yesterday. There are many ways of writing it. There is a single point in the lattice. So one way of writing it would be like this, that if I consider all points that project down to X and which lie in the window set in the model space, there is only a single lattice point in here. So there's exactly one point that gives this. Okay. And in this general situation, we have the point that P always has an asymptotic density. So that is the theorem by Hoff and Schlotman. So this was, Jens had this yesterday also, wrote this down. They proved it independently. So no, we could allow non-connected W. So the formula, which Jens wrote out was for the number of points. I'm going to write something here, but if you take, count the number of points in some ball. So a ball of radius T and then this is, and divide by the volume of the ball, then this tends to some number. No, limit is equal to some number, theta. But I will state it slightly more generally now. So instead of taking just an expanding ball, let's take a somewhat more general expanding set. So replace this by T to the T times some fixed set D. And here, we take the volume of this. And here, this actually holds for an arbitrary. So let D be a fixed bounded subset of our D with a boundary of measures zero. So D is Jordan measurable. And for this to make sense, we have to assume also that, well, I guess that D has positive volume or equivalently D has non-empty interior. And then the statement is that this holds. And furthermore, we can tell what the asymptotic density is. We can tell what theta P is in terms of the lattice and the window set. So this holds with theta equal to the volume of the window set divided by the volume of, or the co-volume of the lattice. When I will have to do this, I will write volume for Lebesgue measure, but it depends on the context which dimension we are in. So here, it's m-dimensional Lebesgue measure. And here, it is n-dimensional Lebesgue measure. And in order to connect with the statement which Jens had yesterday, just note that in general, this would be the volume, we introduced the Lebesgue measure on the set V. And we had V divided by L intersected with V. But as Jens pointed out, if A is equal to RM as we are assuming, then we get just this. Okay, so now I wanted to outline the proof and see how it comes from Vile liquid distribution. Maybe I should have drawn a picture a bit earlier, perhaps. So here's the situation we have. It's hard to draw high-dimensional pictures, but you can imagine that the whole space, RD, lies along this. And here is the whole space, RM. And then we have some lattice. It's only, I can only draw a two-dimensional lattice. No, larger, I think. And why not? I should do as Jens did, maybe. So this is the point set L. And then we have some window. And maybe, let me put, let's say that the window is here. Let's say that, I don't know what to do. What am I doing? I don't know, let's continue the lattice a bit. Okay, and let's assume that the window is maybe here. And then we are looking at all the lattice points that project to the window. So we are actually counting lattice points. I mean, every such lattice point gives rise to a point in P by projection down to RD. So we are actually, what this is counting is the number of lattice points in a long, but in a certain region which is long in some directions, I mean, which is wide in some directions, but can be arbitrarily thin in certain other directions. So this region is nicely expanded and big in RD, but in the RM it is as small as you choose. The window set could be really small here. So this is a standard kind of lattice counting problem. I mean, it's very classical to count the number of points in a big ball and then you are interested in precise error terms also. But it's like a slight variant. Now we are counting it in a region which is long and big in certain directions, but really thin in other directions. So if I try starting the proof, what we want to prove is here I rewrite it only slightly. We want to prove that the limit as T goes to infinity of the number of lattice points inside the window times TD. So in the other way, so TD times the window. This number of lattice points is asymptotic to the volume again of TD. So it's very close to this relation. The only thing is I'm counting lattice points instead of counting points in P. And of course, when I say that those two are equivalent, I am using this in the activity assumption that there are never two lattice points that give the same point in P. And now let's look at this guy here. What is this? I want to rewrite it slightly and focus instead on the question, let's walk through the lattice points and see when it's a lattice point registered. No, no, no, no. Sorry, I want to focus on this multi-dimensional flow which we are kind of in the RD component. We are moving very widely in every direction. So I want to focus on the point lying in TD and ask when is, so point lying along the RD component and ask when does such a point happen to register as a count here? So this is equal to, we have this normalizing factor and then we are counting the number of points X in TD, such that, and if you look at the condition for such a point X, we have to, the condition is that X times some point in the window should lie in L. So in other words, if I subtract the point in the window, it implies that X comma zero, now this is a point in RN, sorry for the confusion, that this lies in minus the window set, but the window set is in the, maybe minus zero times the window set plus L. And then when I write the pair like this, I mean this is the RD component, this is the RM component. So all I did was I, instead of, yeah, so for each lattice point, we are adding the minus the window set to it. So for each, so instead of kind of looking at the window, I'm looking at the minus the window, but I added that to each lattice point. So here is lattice point plus minus the window. Here is also minus the window. And here is also lattice point plus minus the window. Now that was exactly below. So minus the window. And here lattice point plus minus the window. So we have, to each lattice point, we have now added the window, the minus the window. And the question now is, as I walk along the RD, how many times do I happen to hit? Do I go through such a, such a window? So to say, this is what I'm counting. I'm walking around in RD and I want to count how many times do I hit this M dimensional set, which is kind of transversal to the flow. And I'm going out all the way through TD, whatever TD is, it goes out maybe here. TD, it's a large big set in RD. Okay, so now I want to see this as some kind of flow in the torus that I get by modding out by the lattice L. And I want to get hold of this limit by using the fact that this D dimensional flow equidistributes this D dimensional orbit equidistributes in the torus. And then I want to see this as an ergodic average. But here I'm counting something. So it's not really an ergodic average, an ergodic for something that you can get by the, as an equidistribution. There you want to count how many times do we hit some open set, for instance. And this is, so what we do is we thicken, we thicken these windows a little bit in the RD directions. So then we will be counting some volume instead. So let's thicken it by some amount, small amount eta in the RD directions. And here we have to be, we have to be careful to take eta so small that the different thickenings are this, otherwise disjoint, otherwise the count will not be exactly equal to the volume that we want. So we take eta small, so such, so that if I now take the eta ball in RD times the window set, or it should be minus the windows, ah, it can be, that doesn't matter, times the window set, and add the lattice point, that should always be disjoint from any other such neighborhood of a lattice point. So this, these two are always disjoint when m and m prime are different lattice points. So this is empty for all m not equal to m prime in the lattice. Okay, so we have this picture. I have thickened this into an open set. Now this is an open set in RN, and I thickened it by multiplying by ball of diameter two eta, if you like. So two eta each point here. And they will be disjoint by choice of eta. Yeah, now these regions are open in RN. These regions, they are open in RN. I mean, okay, window set is, you can think of window set as an open set. The boundary smash is zero. Sorry. Okay, but of course we won't get this count on the nose because there is this, whenever I'm near, if I happen to have some such window set very near the boundary of TD, then there can be a little bit of, the part of this region could be inside and part of it could be outside of TD. So I can't, can never get it on the nose. So all I can do is estimate it from above and below. So to handle that, I will do the estimation from above. We introduce a little thickening of D. So also set D eta plus, let this be equal to the eta thickening of D. So D plus B, the ball of radius eta. This is pointwise addition. So for every point in D I consider also it's eta neighborhood and I get the slightly larger set. This is the eta thickening of D. And note that if I assume that T is larger than one as I may, then if I take T times D, and take the eta thickening of that, this is a subset of T times the eta thickening. So it's convenient somehow to have a completely fixed set. I will use this fixed set and take it times T and that will overshoot what I need, so to say. So now finally we can get a good upper bound for this thing. I'm going very slowly, I feel, but it's morning and I may not be completely awake yet. I hope you're not too, don't find it too slow. Okay, so finally we conclude that. So I'm going to replace that count by a certain volume. And so I keep this factor, volume of TD. I save some space here for an extra correction factor and then I'm now going to count the volume inside the TD or rather TD eta. I'm going to count the volume of points which fall inside these neighborhood, these lattice neighborhoods. So that's what I'm going to count and I write it as an integral. So I go through all the points in TD eta thickened and I register, so this is indicator function giving one whenever the following holds. I register this point whenever we have, whenever that point falls inside this neighborhood of the lattice point. So whenever it falls inside this plus the lattice and volume. And clearly now for every lattice point that I register in this count, since I have thickened the D a little bit, I will get the full eta ball registered here. So I will overshoot if I also divide it by the volume of the eta ball, I get something that is larger than what I needed. So okay, I'm going through this very rather carefully but as I said, this is illustrating a standard technique going from an equidistribution of some orbits of a flow to counting points into the points. And that technique has been applied over and over again to get interesting number theoretical results. Okay, and now the final thing is to view this as something taking place on the torus that I get by taking Rn modulo n, the Rn modulo l. So let pi be the projection map from Rn to that torus. And then note that this, sorry, no. This is equivalent to saying that when I project, since I have this plus l, this set is actually a set living on the torus. And what this is saying is that when I project X comma zero to the torus, it falls inside the projection of this set. I can write it as just projection of minus B eta D times W. And this is an open set in the torus. I assume that eta is, that W is open so that I am allowed to say open here. I can assume so because boundary measures zero. Okay. So now this is really an ergodic average on the torus for this D dimensional flow. So I have a D dimensional flow going in all the directions of Rd. And so now we have a general equidistribution facts on the torus. So it turns out that this D dimensional flow is in such a direction that it is ergodic and hence uniquely ergodic. So all this business, maybe I should go through it more carefully, but there is no time then. So I just say this, that the assumption we started from that the projection of the lattice is equal to all of Rm. This implies that this is more or less immediate if you think about what it means that if I take the closure of that point-wise sum, it is equal to all of Rn. And if I interpret this on the torus, it says that if I take the Rd flow, so I write it like this, and I consider it on the torus, this is dense in the torus. So the closure is equal to all of the torus. And because I have such a denseness, the flow is also uniquely ergodic. So if I take any invariant limit measure, must be the Lebesgue volume, normalized Lebesgue volume measure here. So I don't say more, I just say now that now we have equidistribution. So this ergodic average tends to the volume of this divided by the volume of the full torus. So this, as T goes to infinity, tends to, and I have to move around, maybe I removed the statement, you have it in your notes anyway. So then I can write it here. It tends to, well, this is a factor that just remains one over the d-dimensional volume of this eta ball times. And so it's not really an ergodic average. It would be if I divide by the volume of this thing. So in order to compensate for that, I have to have this compensating factor, volume of d eta plus divided by volume of d. Really, that should be a T here. But that T cancels in the numerator cancels against the denominator. So, and then now I have volume of T d eta plus. And it's really an ergodic average. And by equidistribution, it tends to the volume of this set divided by the volume of the full torus. And it was maybe minus, but that doesn't matter. And now this volume here, I can remove pi because this projection is injective. Why? It's that is equivalent to the assumption I made here. So this is equal to the Lebesgue measure in Rn of this thing. And then this is a product of the volume of this ball which cancels against that thing and the m-dimensional volume of m. So I'm left with this adjusting factor times and this cancels, so it's the volume of that thing divided by the volume of the torus. And now I'm more or less done. So of course, this factor is slightly larger than the one. But the thing is by choosing it as small, I can make it as small as I like because d is the order measurable. So what follows from this discussion? I had one inequality, so the conclusion is that the lim sup as t goes to infinity of what I'm interested in is less than or equal to the thing up here, this thing. But now I let eta go to zero in this statement and this thing goes to one because d is Jordan measurable. So now let eta go to zero and then we conclude what we want. We conclude that the lim sup of what we are interested in is less than or equal to the stated limit. This is exactly the stated theta if you look back into the statement of the theory. So lim sup is good. And then to treat the lim in, of course you do the thinning instead. In order to get an equality from the other direction then you take the eta thinning of d which would be the set of points which have a full eta neighborhood inside d. But again we will have such a fact that the volume of that thinning can be made arbitrarily close to the volume of d. And we will get again that the lim in is also okay. So I spent really a long time on this. I hope it can be helpful for at least some of you. I'm happy to discuss if there are some technicalities here that you want to. In almost the whole argument eta is fixed. And I keep eta fixed all the time until I have reached this conclusion that the lim sup at t goes to infinity. Ah, sorry. Ah, the fact that the projection is injective. There is a slight thing that I wanted to mention and it could be that the lattice is so aligned that actually there are several lattice points along zero times rm. In that case I would first, and then if the window is large I'm dead already for eta equals one. So in that case I have to first partition the window into smaller parts. Ah, but I have prejudiced this somehow. Ah, but this is a statement really in taking a fundamental domain for the tourists and looking at it. This can be done. Maybe we can discuss... Yeah, for a generic lattice there will be points arbitrarily close to rm. But this is really a statement on the tourists so it's no problem and we can discuss the technicalities. I guess I can't say anything that is really the correct thing to say but it is no problem. It's something that... Okay, this will make sense. To verify this it suffices to take m' equals the origin. If it's also for m' equals the origin and all other m's then I'm done because of periodicity. And then it's easy to... There are only a finite number of lattice points that are at all relevant somehow. You can restrict to a finite number of lattice points. I hope that makes sense. Okay, so now in the next 20 minutes I wanted to speak about visible points in p. So, remember that... I think Jens did this yesterday. We are standing at the origin and we are looking out at our points at p. And then clearly we will not see all the points because some... It may happen that we won't see all the points in p because some points may be completely shadowed. I mean hidden. They are along the same ray from the origin. And we call the points that we can see. We call them visible. Of course, if the lattice is in really generic position then I get some points at p. Then I won't have any difference between visible points and other points. It won't happen. It's a non-generic phenomenon that a point is hidden from the origin by some other points. But this happens very much for some of the classical examples of quasi-crystals. So, for instance, in the penrose tiling if you take the vertex set of the penrose tiling as Jens described yesterday then there are invisible points. So there is a real distinction between visible points and the full set. And the same holds for many other of the quasi-crystals that you get from this number field construction, which Jens did yesterday. So we define p hat to be the set of visible points. So it's a set of all points in y such that there is no point ty with t less between 0 and 1, hitting it. So it should be true that ty is never in p for all t between 0 and 1. This is the subset of visible points. I should... And I note that the origin is never in p even if the origin is in... It's never in p hat even if it is in p even if it may be in p. Okay. So now it's clearly an interesting question to ask what is the density of p hat? Can we get hold of it? So... And there's a recent paper by Bakke, Götze, Huck and Czakobi where they investigate this and they investigate also the statistics of directions the thing that Jens spoke a few... Wednesday I think. There are several such sets. p and the corresponding set of visible points. So I should have looked if this has now appeared. I'm going to refer to Archive 2014 but has it appeared, do you know? Okay. Anyway, you may look if you want to see pictures and some statistics and so on. So... Okay. Okay. So you will find it. It has appeared. Okay. But anyway, I don't have that much time but let me state a few theorems and try to give the main idea behind them. So you can actually prove that also this point set p hat always has an asymptotic density. And this is... Yeah, it's a paper by Jens and me from last year. So the statement is this. There exists... I mean p is always fixed and so also p hat is fixed. Now the statement is that there exists some kappa in the... it's strictly positive and it's less than or equal to 1 such that... again for any d... for any Jordan set d. So d is bounded subset of rd volume of the boundary is equal to zero. We have limit as t goes to infinity of the number of points in p hat intersected with td divided by the volume of td. This is equal to the original density but multiplied by kappa. So just as an example let me point out if p is zd. Okay, I should have said this earlier to illustrate just what we mean by visible points. Then p hat is the set of primitive lattice vectors. We often write z hat d for that set and it's the set of all integer vectors such that the greatest common divisor of all the coordinates is equal to 1. And in this case it is a well-known fact that the density of z hat d is equal to 1 over the Riemann zeta function. So then in this case the density of p is clearly 1 and the density of p hat or zd hat is equal to... well it's the product of all prime numbers so p runs through all prime numbers and then 1 minus 1 over p to the d. So this is 1 over zeta function of d. And the intuitive reason for this is kind of clear. For each prime you have to remove any point M which where all the coordinates is divisible by p and the density that you are then removing is 1 over p to the d. So you keep 1 minus 1 over p to the d and you do this for each prime and it's kind of independent conditions so this is what you should get. And it's easy to make this rigorous by doing counting and great Chinese remainder theory. Here we don't have really an explicit formula for kappa of p but as you will see we have somehow we have a... some... it should... there is one should be able to maybe say something about it. Maybe we won't do so but... okay. So the fun thing about this theorem is that it actually uses the Siegel-Wietz result and it uses this space of Cauchy crystals that Jens spoke about yesterday. So although this is a question only discussing the fixed points at p when we prove it we need to introduce this larger space of all Cauchy crystals associated to p. So I have to remind you of this space of Cauchy crystals associated to p. So again this refers back to Jens lecture yesterday I will be a little bit brief now but... so... we had this lattice L and we take some G in SL... in G tilde G tilde is equal to SLNR we take it so that it gives the lattice L so G is just the representative of the fixed lattice L the lattice L which is used to construct p. So such that L is equal to ZNG and in Jens lecture yesterday we also had the delta here but let me just assume that the dense that the co-volume of L is 1 we can do so after re-scaling so in like we are assuming delta equals 1 if you refer back to Jens lecture and then from this G using this G Jens defined an embedding into the homogenous space gamma tilde slash G tilde and then using Ratter Lake with this distribution or densities it follows that for any given such G there exists some closed connected subgroup HG I'm just... so it's a closed connected subgroup of G tilde and it has the property that it intersects gamma tilde in a lattice gamma tilde intersects with HG is a lattice in HG remember that gamma tilde is SLN Z so then we have a homogenous space it's HG modular this lattice so HG modular the lattice gamma tilde intersected with HG and this space can also be identified with just this subset of a certain subset of G tilde slash gamma tilde so this is a subset a nicely embedded sub manifold inside G tilde slash gamma tilde and we have with this space is also provided with a probability measure because this is a lattice there is a unique R measure on HG which gives an invariant probability measure on X and as the answer I call this mu G yeah and now having done all this we have our space of random quasi-crystals random or the space of quasi-crystals associated to the fixed quasi-crystal P so by a random quasi-crystal we mean the quasi-crystal obtained by using the window set W and then taking the lattice not Z and G as from start but we insert an H here so it's Z and HG and this is for the point H or if you like gamma tilde H random in X according to X provided with its invariant probability measure so this is our set of quasi-crystals and this is what if I take this point H at random this is what I mean by a random quasi-crystal just as yesterday we were speaking about random at random lattice now we have a natural notion of a random quasi-crystal in RD so this is also if you refer back to the more general framework with random point processes and so on this is the answer the XI so we will have so this is the XI that we get in the limit and for this XI what Jens outlined yesterday is a proof that XI T tends in distribution to XI remember this is the fixed point set P randomly rotated and then dilated dilated so just to remind you of the notation and now let me state the Siegel-Wietz formula for this so in my lecture yesterday I outlined how Siegel-Wietz general Siegel type formula should follow if you have certain conditions and if you collect various things that we have said today and yesterday you can check that really we have those conditions so we do obtain a precise Siegel-Wietz formula for this space of run of quasi-crystals so let me state it so this is the Siegel-Wietz formula for this special case for any L1 function on RD if I if I integrate over the space of quasi-crystals and really X now is a we can we have identified X with our set of quasi-crystals through this map if I take a point gamma till the H in X this corresponds to this quasi-crystal if I so I integrate over all quasi-crystals and then I add over all the points in that quasi-crystal so I M runs through all the point in the corresponding quasi-crystal this is the Siegel transform I write it directly here and I have to remove the origin as you remember yesterday and I add over all these points F and then I take the average or expected value of this sum and the answer is that this tends to just the volume integral of the Lebesgue integral of F times the appropriate number and this number is the density of P in this case okay we have this but now using exactly the same argument one can show that the same thing must hold if I instead of P here I use P hat I mean for each such for each quasi-crystal in my space of quasi-crystal I can consider the subset of visible points okay and let's do that so and there is some constant kappa P and it's something between 0 and 1 such that if I now integrate again I do the same integral I write it out again I could do it easily by just taking hat here I want to maybe have them both on the on the board to discuss so we add over all the points in same random quasi-crystal but now I take its subset of visible points and then I don't have to remove the origin because the origin is never there and I do the same thing I add over F and then average over the space and this is equal to kappa P theta P volume integral of okay so maybe to avoid so okay actually I'm not going to get time to say much more so really this is the same constant so I don't have to so in order to have the statement here I write it like this kappa P is the same constant as in theorem 2 but now I will just say in words try to give you some idea of what is happening the way we prove these two theorems is we first prove theorem 3 and theorem 3 can be proved I mean this part has already been proved if you trust everything but the second part the proof of the second part is just the same as I outlined yesterday morning because this point set can again be seen to be this random point set can be seen to be an SLDR invariant point set whenever you have a random point set if you take visible points if you apply hats to it you again get the SLDR invariant point set I mean the operation of removing all the invisible points that is invariant under the action of SLDR clearly so the argument goes through and if you remember there was some technicality that we needed an upper bound but that upper bound is already provided from here because the visible points is always a subset of the full set so there is no real problem the argument carries through and we get this so the conclusion is that if you are now trying to outline the proof of theorem 2 and theorem 3 very quickly we first prove theorem 3 and the conclusion is that there exists some constant kappa p such that this holds and then it requires quite bit more work to prove theorem 2 not that those two theorems are quite different statements theorem 3 says that if I take a fixed set I mean if I take instead of a function I take a Borel subset of rd and then I take f to be its characteristic function then this statement in theorem 3 says that for a fixed set if I run through the whole space of cosy crystals the expected number of points in the set is equal to this this kind of density whereas in theorem 2 we are instead looking just as one fixed cosy crystal we don't state anything about the family only for one fixed cosy crystal and we are looking at larger and larger subsets of that counting the number of visible points to different statements but the way we but still using the techniques of spherical averages so I would have enjoyed saying outlining the proof it's you can use the technique of spherical averages to if you go backwards so to say in what Jens showed you in his lecture this Wednesday you can still get the information from the Siegelwitz down to this individual set p but it is I think I end there